[GAP Forum] Question about the groups C91:C3 in AllGroups(273).

[Nilo de Roock]

Dear GAP users,

I am using GAP 4.4.7, on a PC with Windows XP.

For my research I need to know the difference between AllGroups(273)[3] and
AllGroups(273)[4] and also how I can construct them other than using
AllGroups. Sofar I can only construct one and I can't find any group
property which is different between the two.

My first question is about constructing C91:C3. AllGroups(273) yields.

["C13 x (C7 : C3)",
"C7 x (C13 : C3)",
"C91 : C3",
"C91 : C3",
"C273" ]

I have been able to construct one instance of C91:C3 through

gap> G:=CyclicGroup(IsPermGroup,3);
Group([ (1,2,3) ])
gap> N:=CyclicGroup(91);
<pc group of size 91 with 2 generators>
gap> A:=AutomorphismGroup(N);
<group with 2 generators>
gap> f:=GroupHomomorphismByImages(G,A,[Elements(G)[2]],[Elements(A)[39]]);
[ (1,2,3) ] -> [ Pcgs([ f1, f2 ]) -> [ f1^4*f2^10, f2^9 ] ]
gap> NG:=SemiDirectProduct(G,f,N);
gap> NG:=SemidirectProduct(G,f,N);
<pc group of size 273 with 3 generators>
gap> StructureDescription(NG);
"C91 : C3"

How can I construct the other C91:C3 ? ( I have tried mappings to other
elements in A ( C12 x C6
) but none of them eventually results to a different C91:C3 in the
SemidirectProduct.



My second question is about the difference is between AllGroups(273)[3] and
AllGroups(273)[4], they seem isomorphic to me. I have tested various
properties and I can't figure out the difference.

About AllGroups(273)[3] and AllGroups(273)[4]. I name the groups G1 and
G2...
gap> G1:=AllGroups(273)[3];
<pc group of size 273 with 3 generators>
gap> G2:=AllGroups(273)[4];
<pc group of size 273 with 3 generators>

If the subgroups of G1, G2 are the same and the orders of the
elements... then where is the difference? I can't find it in GAP.

The following commands illustrate some significant similarities
between the groups.

gap> StructureDescription(G1);
"C91 : C3"
gap> StructureDescription(G2);
"C91 : C3"
gap> List(ConjugacyClassesSubgroups(G1),Representative);
[ Group([ ]), C3, Group([ f2 ]), Group([ f3 ]), Group([ f1, f2 ]),
Group([ f3, f1 ]), C91, Group([ f3, f2, f1 ]) ]
gap>
List(List(ConjugacyClassesSubgroups(G1),Representative),StructureDescript$
gap> $cription);
[ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ]
gap>
List(List(ConjugacyClassesSubgroups(G2),Representative),StructureDescript$
gap> $cription);
[ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ]
gap> $cription);
[ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ]
gap> Sum(List(Elements(G1),Order));
7297
gap> Sum(List(Elements(G2),Order));
7297

Thanks on beforehand for any hints on this particular issue.

-- 
met vriendelijke groet / kind regards,
nilo de roock